Parity Conservation in the weak (beta decay) interaction.
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The parity operation
The parity operation involves the transformation
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x → −x y → −y z → −z
r → r θ → π −θ( ) φ → φ + π( )
In rectangular coordinates --
In spherical polar coordinates --
In quantum mechanics
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ˆ Π ψ x,y,z( ) = ±1ψ −x,−y,−z( )
For states of definite (unique & constant) parity -
If the parity operator commutes with hamiltonian -
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ˆ Π , ˆ H [ ] = 0 ⇒ The parity is a “constant of the motion”
Stationary states must be states of constant parity
e.g., ground state of 2H is s (l=0) + small d (l=2)
In quantum mechanics
To test parity conservation - - Devise an experiment that could be done: (a) In one configuration (b) In a parity “reflected” configuration- If both experiments give the “same” results, parity is conserved -- it is a good symmetry.
Parity operations --
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ˆ Π E = E ˆ Π r r ⋅
r r ( ) =
r r ⋅
r r ( )
Parity operation on a scaler quantity -
Parity operation on a polar vector quantity -
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ˆ Π r r = −
r r ˆ Π
r p = −
r p
Parity operation on a axial vector quantity -
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ˆ Π r L = ˆ Π
r r ×
r p ( ) = −
r r × −
r p ( ) = −
r L
Parity operation on a pseudoscaler quantity -
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ˆ Π r p ⋅
r L ( ) = −
r p ⋅
r L ( )
If parity is a good symmetry…• The decay should be the same whether the process
is parity-reflected or not.
• In the hamiltonian, V must not contain terms that are pseudoscaler.
• If a pseudoscaler dependence is observed - parity symmetry is violated in that process - parity is therefore not conserved.
T.D. Lee and C. N. Yang, Phys. Rev. 104, 254 (1956).
http://publish.aps.org/ puzzle
Parity experiments (Lee & Yang)
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rp ⋅
r I = 0
Parity reflectedOriginal
Look at the angular distribution of decay particle (e.g., red particle). If this is symmetric above/below the mid-plane, then --
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rp
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rI
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rp
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rI
If parity is a good symmetry…
• The the decay intensity should not depend on .
• If there is a dependence on and parity is not conserved in beta decay.
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rp ⋅
r I = 0
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rp ⋅
r I ( )
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rp ⋅
r I ≠ 0
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rp ⋅
r I ( )
Discovery of parity non-conservation (Wu, et al.)
Consider the decay of 60Co
Conclusion: G-T, allowed
C. S. Wu, et al., Phys. Rev 105, 1413 (1957)
http://publish.aps.org/
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Hβ − = −
v
c
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ν
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Hν =1
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β−
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ν €
Hβ − = −
v
c
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β−
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Hν =1
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ν
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Hν = −1
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ν
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Hν = −1
Not observedMeasure
Trecoil
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GT : ΔI =1
A :1−1
3
v
ccosθ
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F : I = 0 → I = 0
V :1+v
ccosθ
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Hν =1
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Hβ − = −
v
c
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GT : ΔI =1
A :1−1
3
v
ccosθ
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F : I = 0 → I = 0
V :1+v
ccosθ
Conclusions
GT is an axial-vector F is a vector
Violates parity Conserves parity
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VN −N = Vs + Vw
ImplicationsInside the nucleus, the N-N interaction is
Conserves parity
Can violate parity
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ψ =ψs + Fψ w ; F ≈10−7
The nuclear state functions are a superposition
Nuclear spectroscopy not affected by Vw
Generalized β-decay
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H = g ψ p* γ μψ n( ) ψ e
*γ μψ ν( ) + h.c. = gV
H = g ψ p* γ μγ 5ψ n( ) ψ e
*γ μγ 5ψ ν( ) + h.c. = gA
The hamiltonian for the vector and axial-vector weak interaction is formulated in Dirac notation as --
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H ≈ g CVV + CA A( )
Or a linear combination of these two --
Generalized β-decay
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H = g Ci ψ p* Oiψ n( ) ψ e
*Oi 1+ γ5( )ψν[ ]i=A,V
∑ + h.c.
OV = γμ ; OV = γμ γ5
The generalized hamiltonian for the weak interaction that includes parity violation and a two-component neutrino theory is --
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g CA CVEmpirically, we need to find -- and
Study: n p (mixed F and GT), and: 14O 14N* (I=0 I=0; pure F)
Generalized β-decay14O 14N* (pure F)
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ft =2π 3h7 log2
m5c4 M F
2g2CV
2( )
g2CV2
( ) =1.4029 ± 0.0022 ×10−49 erg cm3
n p (mixed F and GT)
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ft =2π 3h7 log2
g2m5c4 CV2
( ) M F
2+ CA
2( ) MGT
2 ⎡ ⎣ ⎢
⎤ ⎦ ⎥
Generalized β-decay
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2CV2
CV2 + 3CA
2= 0.3566 →
CV2
CA2
=1.53
Assuming simple (reasonable) values for the square of the matrix elements, we can get (by taking the ratio of the two ft values --
Experiment shows that CV and CA have opposite signs.
Universal Fermi InteractionIn general, the fundamental weak interaction is of the form --
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H ∝ g V − A( )
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n → p + β − + ν e
μ − → e− + ν e + ν μ
π − → μ − + ν μ
Λo → π − + p
semi-leptonic weak decay
pure-leptonic weak decay
semi-leptonic weak decay
Pure hadronic weak decay
All follow the (V-A) weak decay. (c.f. Feynman’s CVC)
Universal Fermi InteractionIn general, the fundamental weak interaction is of the form --
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H ∝ g V − A( )
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μ−→ e− + ν e + ν μ pure-leptonic weak decay
The Triumf Weak Interaction Symmetry Test - TWIST
BUT -- is it really that way - absolutely?
How would you proceed to test it?
Other symmetriesCharge symmetry - C
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n → p + β − + ν e ⇒ n → p + β + + ν e C
All vectors unchanged
Time symmetry - T
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n → p + β − + ν e ⇒ n ← p + β − + ν e
ν e + n → p + β −
T
All time-vectors changed (opposite)
(Inverse β-decay)
Symmetries in weak decay at rest
Note helicities of neutrinos
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rs π = 0
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μ−
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νμ
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μ+€
+
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⇒
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⇐ CP
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−€
ν μ
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μ−
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ν μ€
−
Symmetries in weak decay at rest
Note helicities of neutrinos
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rs π = 0
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+
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νμ
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⇒CP
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−
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μ−
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ν μ
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μ+
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⇒
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ν μ€
−€
μ−
Conclusions1. Parity is not a good symmetry in the weak
interaction. (P)
2. Charge conjugation is not a good symmetry in the weak interaction. (C)
3. The product operation is a good symmetry in the weak interaction. (CP) - except in the kaon system!
4. Time symmetry is a good symmetry in the weak interaction. (T)
5. The triple product operation is also a good symmetry in the weak interaction. (CPT)
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